The general Brannan coefficient conjecture and Watson's lemma

TL;DR

This paper investigates the Brannan coefficient conjecture for certain complex functions, using hypergeometric integrals and Watson's lemma to verify the conjecture for specific parameter ranges and odd integers.

Contribution

It extends the verification of Brannan's conjecture to broader parameters and odd integers by employing integral representations and numerical minimization techniques.

Findings

Conjecture holds for , eta , |\u2202arg(\u03c9)| , and odd n=5,7,9.

Utilizes hypergeometric integrals and Watson-type approximations.

Reduces the problem to numerical evaluation of minima of explicit functions.

Abstract

The coefficients $A_n(\alpha,\beta,\omega)$ in the Maclaurin expansion $(1+\omega z)^{\alpha}(1-z)^{-\beta}= \sum_{n=0}^{\infty} A_n(\alpha,\beta,\omega)z^n$ are studied, where $\omega,z \in \mathbb{C}$ with $|z| < |\omega|=1$, and $\alpha,\beta \in (0,1]$. In 1973 Brannan conjectured that $|A_n(\alpha,\beta,\omega)|\le A_n(\alpha,\beta,1)$ for each positive odd integer $n$, and showed it is true for $n=3$. This has recently been proven for all odd integers $n\ge5$ by a number of authors in aggregate for the special case $\beta=1$. In this paper hypergeometric integral representations and Watson-type approximations are utilised, from which the general problem is reduced to numerically evaluating the minima of certain simple, explicit, slowly-varying functions over compact domains. From the positivity of these constants it is shown that the conjecture holds for $\alpha, \beta \in (0,1]$, $0 \le |\arg(\omega)| \le \pi-\phi_0$ and $n=5,7,9,\ldots$, where $\phi_0=0.061$.